A022004 -id:A022004 - cp's OEIS frontend

A236508 a(n) = |{0 < k < n-2: p = 2*phi(k) + phi(n-k)/2 - 1, p + 2, p + 6 and prime(p) + 6 are all prime}|, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 2, 1, 3, 2, 2, 0, 2, 3, 1, 2, 1, 3, 3, 2, 2, 1, 1, 1, 3, 0, 2, 3, 2, 1, 3, 0, 2, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 1

Zhi-Wei Sun, Jan 27 2014

nonn

Conjecture: a(n) > 0 for all n > 146.
We have verified this for n up to 52000.
The conjecture implies that there are infinitely many prime triples {p, p + 2, p + 6} with {prime(p), prime(p) + 6} a sexy prime pair. See A236509 for such primes p.

			a(13) = 1 since 2*phi(3) + phi(10)/2 - 1 = 5, 5 + 2 = 7, 5 + 6 = 11 and prime(5) + 6 = 11 + 6 = 17 are all prime.
a(244) = 1 since 2*phi(153) + phi(244-153)/2 - 1 = 2*96 + 72/2 - 1 = 227, 227 + 2 = 229, 227 + 6 = 233 and prime(227) + 6 = 1433 + 6 = 1439 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A022004, A023201, A046117, A236464, A236472, A236509.

p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[n+6]&&PrimeQ[Prime[n]+6]
f[n_,k_]:=2*EulerPhi[k]+EulerPhi[n-k]/2-1
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

A286891 Initial primes of 6 consecutive primes with 5 consecutive gaps 10, 8, 6, 4, 2.

Original entry on oeis.org

41203, 556243, 576193, 715849, 752263, 859249, 891799, 1107763, 1191079, 1201999, 1210369, 1510189, 1601599, 1893163, 2530963, 2678719, 2881243, 3257689, 3431479, 3545263, 3792949, 3804919, 4041109, 4479463, 4867309

Offset: 1

Muniru A Asiru, Jul 22 2017

nonn

All terms = {13,19} mod 30.

For initial primes of 6 consecutive primes with consecutive gaps 2, 4, 6, 8, 10 see A190817.

			Prime(4313..4318) = {41203, 41213, 41221, 41227, 41231, 41233} and 41203 + 10 = 41213, 41213 + 8 = 41221, 41221 + 6 = 41227, 41227 + 4 = 41231, 41231 + 2 = 41233.
Also, prime(68287..68292) = {859249, 859259, 859267, 859273, 859277, 859279} and 859249 + 10 = 859259, 859259 + 8 = 859267, 859267 + 6 = 859273, 859273 + 4 = 859277, 859277 + 2 = 859279.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
R. J. Mathar, Table of Prime Gap Constellations

Cf. A078847, A190814, A190817, A190819, A190838, A290161, A290162.

Subsequence of A031928.

P:=Filtered([1..20000000],IsPrime);;  I:=Reversed([2,4,6,8,10]);;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
P2:=List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3],P1[i+4]]);;
P3:=List(Positions(P2,I),i->P[i]);

K:=10^7: # to get all terms <= K.
Primes:=select(isprime,[seq(i,i=3..K+30,2)]): Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1], Primes[t+3]-Primes[t+2], Primes[t+4]-Primes[t+3], Primes[t+5]-Primes[t+4]]=[10,8,6,4,2], [$1..nops(Primes)-5])]; # Muniru A Asiru, Aug 15 2017

Select[Partition[Prime[Range[340000]],6,1],Differences[#]=={10,8,6,4,2}&][[All,1]] (* Harvey P. Dale, Aug 22 2018 *)

A292224 Irregular triangle read by rows. T(n, k) gives the number of admissible k-tuples from the interval of integers [0, 1, ..., n-1] starting with smallest tuple member 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 4, 4, 1, 1, 4, 4, 1, 1, 5, 6, 2, 1, 5, 6, 2, 1, 6, 11, 8, 2, 1, 6, 11, 8, 2, 1, 7, 15, 14, 4, 1, 7, 15, 14, 4, 1, 8, 19, 20, 8, 1, 1, 8, 19, 20, 8, 1, 1, 9, 27, 39, 24, 5, 1, 9, 27, 39, 24, 5, 1, 10, 33, 54, 44, 16, 2, 1, 10, 33, 54, 44, 16, 2, 1, 11, 39, 69, 62, 26, 2, 1, 11, 39, 69, 62, 26, 2

Offset: 1

Wolfdieter Lang, Oct 09 2017

The row lengths are given by A023193 (the rhobar function of Schinzel and Sierpiński called rho* by Hensley and Richards).
This irregular triangle has already been considered by Engelsma, see Table 2, for n=1..56, p. 27.
A k-tuple of integers B_k = [b_1, ..., b_k] with 0 = b_1 < b_2 < ... < b_k <= n-1 is called admissible if for each prime p there exists at least one congruence class modulo p which contains none of the B_k elements. (This corresponds to the alternative definition of Hensley and Richards, p. 378 (*) or Richards, p. 423, 1.5 Definition and (*).) Note that the definition of "admissibility" is translation invariant (see the Note by Richards, p. 424, which is obvious from the translation equivalence of complete residue systems modulo p). Therefore the interval I_n = [0, n-1] of length n has been chosen. The b_1 = 0 choice is conventional. Without this choice other admissible k-tuples are obtained by translation as long as b_k + a < n-1. E.g., for n = 8 and k = 3 the tuple [1, 5, 7] is admissible and a translation of the considered tuple [0, 4, 6].
Only primes p <= k have to be tested to decide on the admissibility of a B_k tuple because for larger k there is always some residue class which contains none of the k members of B_k.
Because p = 2 already forbids even and odd numbers to appear in B_k for k >= 2, one can for the admissibility test eliminate all odd numbers in the chosen I_n. Therefore, only Ieven_n:= [0, 2, ..., 2*floor((n-1)/2)] =: 2*[0, 1, ..., floor((n-1)/2)] need be considered. B_1 = [0] is admissible for all n >= 1.
Because only the interval Ieven_n is of relevance, there will occur repetitions for admissible tuples for n if n = 2*k+1 and n = 2*k+2.
With the set B_k(p) = B_k (mod p) := {0, b_1 (mod p), ..., b_k (mod p)} the criterion for admissibility can be written as p - #(B_k(p)) > 0, for all primes 3 <= p <= k (because there are p congruence classes defined by smallest nonnegative complete residue system [0, 1, ..., p-1]).
Admissible tuples (starting with 0) with least b_k - b_1 = b_k value give rise to prime k-constellations of diameter b_k. E.g., for k = 2 the admissible tuple [0, 4] does not lead to a prime 2-constellation for n >= 5; [0, 6] is out for n >= 7, ... . But there are two prime 3-constellations given by [0, 2, 6] and [0, 4, 6] for n >= 7.
Row sums are in A292225, that is, total number of admissible tuples starting with 0 from the interval I_n = [0, n-1].

			The irregular triangle begins:
n\k   1  2  3  4  5  6  7 ...
1:    1
2:    1
3:    1  1
4:    1  1
5:    1  2
6:    1  2
7:    1  3  2
8:    1  3  2
9:    1  4  4  1
10:   1  4  4  1
11:   1  5  6  2
12:   1  5  6  2
13:   1  6 11  8  2
14:   1  6 11  8  2
15:   1  7 15 14  4
16:   1  7 15 14  4
17:   1  8 19 20  8  1
18:   1  8 19 20  8  1
19:   1  9 27 39 24  5
20:   1  9 27 39 24  5
21:   1 10 33 54 44 16  2
22:   1 10 33 54 44 16  2
23:   1 11 39 69 62 26  2
24:   1 11 39 69 62 26  2
...
The first admissible k-tuples are (blanks within a tuple are here omitted):
n\k  1                2                                  3                       4  ...
1:  [0]
2:  [0]
3:  [0]  [0,2]
4:  [0]  [0,2]
5:  [0]  [[0,2], [0,4]]
6:  [0]  [[0,2], [0,4]]
7:  [0]  [[0,2], [0,4], [0,6]]          [[0,2,6], [0,4,6]]
8:  [0]  [[0,2], [0,4], [0,6]]          [[0,2,6], [0,4,6]]
9:  [0]  [[0,2], [0,4], [0,6], [0,8]]   [[0,2,6], [0,2,8], [0,4,6], [0,6,8]]  [0,2,6,8]
10: [0]  [[0,2], [0,4], [0,6], [0,8]]   [[0,2,6], [0,2,8], [0,4,6], [0,6,8]]  [0,2,6,8]
...
The first admissible k-tuples for prime k-constellations are:
n\k  1     2           3                 4                    5                       6  ...
1:  [0]
2:  [0]
3:  [0]  [0,2]
4:  [0]  [0,2]
5:  [0]  [0,2]
6:  [0]  [0,2]
7:  [0]  [0,2]  [[0,2,6], [0,4,6]]
8:  [0]  [0,2]  [[0,2,6], [0,4,6]]
9:  [0]  [0,2]  [[0,2,6], [0,4,6]]   [0,2,6,8]
10: [0]  [0,2]  [[0,2,6], [0,4,6]]   [0,2,6,8]
11: [0]  [0,2]  [[0,2,6], [0,4,6]]   [0,2,6,8]
12: [0]  [0,2]  [[0,2,6], [0,4,6]]   [0,2,6,8]
13: [0]  [0,2]  [[0,2,6], [0,4,6]]   [0,2,6,8]   [[0,2,6,8,12],[0,4,6,10,12]]
14: [0]  [0,2]  [[0,2,6], [0,4,6]]   [0,2,6,8]   [[0,2,6,8,12],[0,4,6,10,12]]
15: [0]  [0,2]  [[0,2,6], [0,4,6]]   [0,2,6,8]   [[0,2,6,8,12],[0,4,6,10,12]]
16: [0]  [0,2]  [[0,2,6], [0,4,6]]   [0,2,6,8]   [[0,2,6,8,12],[0,4,6,10,12]]
17: [0]  [0,2]  [[0,2,6], [0,4,6]]   [0,2,6,8]   [[0,2,6,8,12],[0,4,6,10,12]] [0,4,6,10,12,16]
18: [0]  [0,2]  [[0,2,6], [0,4,6]]   [0,2,6,8]   [[0,2,6,8,12],[0,4,6,10,12]] [0,4,6,10,12,16]
...
-----------------------------------------------------------------------------------------------
T(7, 3) = 2 because Ieven_n = [0, 2, 4, 6], and the only admissible 3-tuples from this interval are [0, 2, 6] and [0, 4, 6]. For example, [0, 2, 4] is excluded because the set B_3 (mod 3) = {0, 1, 2}, thus #{0, 1, 2} = 3 and (p = 3) - 3 = 0, not > 0.
These two admissible 3-tuples both have diameter 6 and stand for prime 3-constellations for all n >= 7: p, p + 2, p + 6, and p, p + 4, p + 6. One of the Hardy-Littlewood conjectures is that there are in both cases infinitely many such prime triples. For the first members of such triples see A022004 and A022005.

Pontus von Brömssen, Table of n, a(n) for n = 1..1296 (rows 1..100)
Thomas J. Engelsma, Permissible Patterns of Primes, September 2009, Table 2, p. 27.
D. Hensley and I. Richards, Primes in intervals, Acta Arith. 25 (1974), pp. 375-391.
Ian Richards, On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem, Bulletin of the American Mathematical Society 80:3 (1974), pp. 419-438.
A. Schinzel, Remarks on the paper 'Sur certaines hypothèses concernant les nombres premiers', Acta Arithmetica 7 (1961), pp. 1-8.
A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arithmetica 4,3 (1958), pp. 185-208, Théorème 1, p. 201; erratum 5 (1958) p. 259.
Wikipedia, Prime k-tuple.

Cf. A020497, A022004, A022005, A023193, A292225.

T(n, k) = number of admissible k-tuples B_k = [0, b_2, ..., b_k] (see the comment above) from the interval of integers Ieven_n:= [0, 2, ..., 2*floor((n-1)/2)].

A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)

Original entry on oeis.org

3, 7, 15, 26, 38, 48, 67, 92, 105, 108, 109, 118, 130, 128, 112, 80, 36, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Dec 19 2002

nonn

The ">n+1" rules out n-tuples like (2,2), which only occurs for the primes 3, 5, 7. All terms from a(19) on equal 0.

An n-tuple (a_1,a_2,...,a_n) is counted iff the partial sums 0, a_1, a_1+a_2, ..., a_1+...+a_n do not contain a complete residue system (mod p) for any prime p.

Eric Weisstein's World of Mathematics, k-Tuple Conjecture

The 26 4-tuples and 38 5-tuples are in A078868 and A078870. Cf. A001359, A008407, A029710, A031924, A022004-A022007, A078852, A078858, A078946-A078969, A020497.

test[tuple_] := Module[{r, sums, i, j}, r=Length[tuple]; sums=Prepend[tuple.Table[If[j>=i, 1, 0], {i, 1, r}, {j, 1, r}], 0]; For[i=1, Prime[i]<=r+1, i++, If[Length[Union[Mod[sums, Prime[i]]]]==Prime[i], Return[False]]]; True]; tuples[0]={{}}; tuples[n_] := tuples[n]=Select[Flatten[Outer[Append, tuples[n-1], {2, 4, 6}, 1], 1], test]; a[n_] := Length[tuples[n]]

Edited by Dean Hickerson, Dec 20 2002

A078948 Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,4,2).

Original entry on oeis.org

29, 59, 269, 1289, 2129, 2789, 5639, 8999, 13679, 14549, 18119, 36779, 62129, 75989, 80669, 83219, 88799, 93479, 113159, 115769, 124769, 132749, 150209, 160079, 163979, 203309, 207509, 223829, 228509, 278489, 282089, 284729, 298679, 312929, 313979, 323369, 337859

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+8, p+12 and p+14 are consecutive primes.

All terms are congruent to 29 (mod 30). - Muniru A Asiru, Sep 04 2017

			59 is in the sequence since 59, 61 = 59 + 2, 67 = 59 + 8, 71 = 59 + 12 and 73 = 59 + 14 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from R. J. Mathar)
R. J. Mathar, Table of Prime Gap Constellations.

Subsequence of A078848. - R. J. Mathar, Feb 10 2013

Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

K:=26*10^7+1;; # to get all terms <= K.
P:=Filtered([1,3..K],IsPrime);;  I:=[2,6,4,2];;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
Q:=List(Positions(List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3]]),I),i->P[i]); # Muniru A Asiru, Sep 04 2017

for i from 1 to 10^5 do if [ithprime(i+1),ithprime(i+2),ithprime(i+3),ithprime(i+4)] = [ithprime(i)+2,ithprime(i)+8,ithprime(i)+12,ithprime(i)+14] then print(ithprime(i)); fi; od;  # Muniru A Asiru, Sep 04 2017

Select[Partition[Prime[Range[26000]],5,1],Differences[#]=={2,6,4,2}&][[;;,1]] (* Harvey P. Dale, Dec 10 2024 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 2 && p3 - p2 == 6 && p4 - p3 == 4 && p5 - p4 == 2, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

A078952 Primes p such that the differences between the 5 consecutive primes starting with p are (4,2,4,6).

Original entry on oeis.org

13, 37, 223, 1087, 1423, 1483, 2683, 4783, 20743, 27733, 29017, 33343, 33613, 35527, 42457, 44263, 45817, 55813, 93487, 108877, 110917, 113143, 118897, 151237, 165703, 187123, 198823, 203653, 205417, 221713, 234187, 234457, 258607, 276817, 284227, 289837, 308923

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+4, p+6, p+10 and p+16 are consecutive primes.

All terms = {7, 13} mod 30. - Muniru A Asiru, Aug 21 2017

			37 is in the sequence since 37, 41 = 37 + 4, 43 = 37 + 6, 47 = 37 + 10 and 53 = 37 + 16 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from R. J. Mathar)
R. J. Mathar, Table of Prime Gap Constellations.

Subsequence of A052378. - R. J. Mathar, Feb 11 2013

Cf. A001223, A022006, A022007, A078866, A078867, A078946-A078971.

K:=2*10^7+1;; # to get all terms <= K.
P:=Filtered([1,3..K],IsPrime);;  I:=[4,2,4,6];;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
P2:=List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3]]);;
P3:=List(Positions(P2,I),i->P[i]); # Muniru A Asiru, Aug 21 2017

for i from 1 to 10^7 do if ithprime(i+1)=ithprime(i)+4 and ithprime(i+2)=ithprime(i)+6 and ithprime(i+3)=ithprime(i)+10 and ithprime(i+4)=ithprime(i)+16 then print(ithprime(i)); fi; od; # Muniru A Asiru, Aug 21 2017

With[{s = Differences@ Prime@ Range[10^5]}, Prime[SequencePosition[s, {4, 2, 4, 6}][[All, 1]]]] (* Michael De Vlieger, Aug 21 2017 *)

lista(nn) = forprime(p=3, nn, if(nextprime(p+1)==p+4 && nextprime(p+5)==p+6 && nextprime(p+7)==p+10 && nextprime(p+11)==p+16, print1(p, ", "))); \\ Altug Alkan, Aug 21 2017

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 4 && p3 - p2 == 2 && p4 - p3 == 4 && p5 - p4 == 6, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

A115786 Smallest prime number p such that p + 2#, p + 3#, ..., p + prime(n)# are all prime, where x# = A034386(x) is the primorial.

Original entry on oeis.org

3, 5, 11, 17, 41, 41, 41, 41, 86351, 86351, 235313357, 729457511, 99445156397, 818113387907, 7986903815771, 29065965967667

Offset: 1

Rick L. Shepherd, Jan 31 2006

Subset of A001359 (lesser of twin primes).

From a(2) = 5 on, also a subset of A022004: first element of prime triples (p, p+2, p+6).-- It could make sense to add a(0) = 2, the smallest prime (with empty further restriction "p + prime(n)# prime for 1 <= n <= 0"). - M. F. Hasler, Apr 29 2015

			a(11) = 235313357 because 235313357, 235313357 + 2, 235313357 + 2*3, 235313357 + 2*3*5, ... and 235313357 + 2*3*5*7*11*13*17*19*23*29*31 are all prime and there is no smaller prime with this property.

C. Rivera, Puzzle 782: Prime-Generating non-polynomials, primepuzzles.net, April 4, 2015
C. Rivera, Puzzle 350. Primes & primorials, primepuzzles.net, 2006

Cf. A001359, A002110, A115785 (for p - p(i)#).

a(12) from Don Reble, Feb 15 2006

More terms from Jens Kruse Andersen, Feb 28 2006

A174858 Primes p of a prime triple (p,p+2,p+6) such that the concatenation p//(p+2)//(p+6) is prime.

Original entry on oeis.org

5, 11, 17, 41, 11171, 16061, 16187, 20897, 29021, 34841, 36011, 39227, 41177, 51341, 55331, 56891, 58907, 63311, 64151, 69191, 77261, 82757, 113021, 122027, 123731, 135461, 151337, 167621, 173291, 174761, 187631, 191447, 195731, 203207, 203381, 225341, 227531

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 31 2010

If p is a d-digit prime of a triple: p*10^(2*d) + (p+2)*10^d + p+6 = (10^(2*d)+10^d+1) * p + 2*(10^d+3) to be a prime.

No such concatenation exists for a 4-digit p: d=4, p*10^8 + (p+2)*10^4 + p+6 = p*(10^8 + 10^4 + 1) + 2*10^4 + 6, coefficients (10^8 + 10^4 + 1) and 2*(10^4 + 3) have both divisor 7.

			(5,7,11) is 1st prime triple, 5711 = prime(752), 5 is 1st term of sequence
(11,13,17) is 2nd prime triple, 111317 = prime(10561), 11 is 2nd term of sequence

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A022004.

Transpose[Select[Partition[Prime[Range[20000]],3,1],Differences[#]=={2,4} && PrimeQ[ FromDigits[Flatten[IntegerDigits/@#]]]&]][[1]] (* Harvey P. Dale, Apr 10 2013 *)

A233434 Primes p in prime triples (p, p+2, p+6) at the end of the maximal gaps in A201598.

Original entry on oeis.org

11, 41, 101, 191, 461, 641, 1091, 1871, 2657, 3251, 6827, 7877, 40427, 47711, 58907, 86111, 171047, 379007, 385391, 553097

Offset: 1

Alexei Kourbatov, Dec 09 2013

nonn

Prime triples (p, p+2, p+6) are one of the two types of densest permissible constellations of 3 primes. Maximal gaps between triples of this type are listed in A201598; see more comments there.

			The gap of 6 between triples starting at p=5 and p=11 is the very first gap, so a(1)=11. The gap of 6 between triples starting at p=11 and p=17 is not a record, so a new term is not added. The gap of 24 between triples starting at p=17 and p=41 is a record gap - larger than any preceding gap; therefore a(2)=41.

Alexei Kourbatov, Table of n, a(n) for n = 1..72
Tony Forbes, Prime k-tuplets
Alexei Kourbatov, Maximal gaps between prime triples
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053, 2013.
Eric W. Weisstein, k-Tuple Conjecture

Cf. A022004, A201598, A201599.

A236509 Primes p with p + 2, p + 6 and prime(p) + 6 all prime.

Original entry on oeis.org

5, 11, 107, 227, 311, 347, 821, 857, 1091, 1607, 1997, 2657, 3527, 4931, 5231, 8087, 8231, 9431, 10331, 11171, 12917, 13691, 13877, 21377, 22271, 24917, 27737, 29567, 32057, 33347, 35591, 36467, 37307, 39227, 42017

Offset: 1

Zhi-Wei Sun, Jan 27 2014

nonn

According to the conjecture in A236508, this sequence should have infinitely many terms.

			a(1) = 5 since 5, 5 + 2 = 7, 5 + 6 = 11 and prime(5) + 6 = 17 are all prime, but 2 + 2 = 4 and 3 + 6 = 9 are both composite.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A022004, A023201, A046117, A236464, A236508.

p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[n+6]&&PrimeQ[Prime[n]+6]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10^6}]

cp's OEIS Frontend

A236508 a(n) = |{0 < k < n-2: p = 2*phi(k) + phi(n-k)/2 - 1, p + 2, p + 6 and prime(p) + 6 are all prime}|, where phi(.) is Euler's totient function.

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A286891 Initial primes of 6 consecutive primes with 5 consecutive gaps 10, 8, 6, 4, 2.

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A292224 Irregular triangle read by rows. T(n, k) gives the number of admissible k-tuples from the interval of integers [0, 1, ..., n-1] starting with smallest tuple member 0.

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A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)

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A078948 Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,4,2).

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A078952 Primes p such that the differences between the 5 consecutive primes starting with p are (4,2,4,6).

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A115786 Smallest prime number p such that p + 2#, p + 3#, ..., p + prime(n)# are all prime, where x# = A034386(x) is the primorial.

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